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Palindromes and their Prime Factors Assignment 3
Palprim5.htm

1. Smallest palindromes with exactly n palindromic prime factors (cwm)
2. Smallest palindromes with exactly n prime factors (cwm)
3. Smallest palindromes with exactly n distinct prime factors
4. Smallest palindromes with exactly n distinct palindromic prime factors
5. Palindromes divided by the palindromic sum of their prime factors (cwm) is a palindrome
6. Palindromes divisible by the palindromic sum of their prime factors(cwm)
7. Palindromes divisible by the palindromic sum of their exclusive palindromic prime factors

1. Smallest palindromes with exactly n palindromic prime factors
    (counted with multiplicity).

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046385

n	palindrome	palindromic prime factors
0	1	-
1	2	2
2	4	2 * 2
3	8	2 * 2 * 2
4	88	2 * 2 * 2 * 11
5	252	2 * 2 * 3 * 3 * 7
6	2772	2 * 2 * 3 * 3 * 7 * 11
7	82728	2 * 2 * 2 * 3 * 3 * 3 * 383
8	2112	2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
9	4224	2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
10	8448	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
11	236989632	2 * 2 * 2 * 2 * 2 * 2 * 3 * 11 * 11 * 101 * 101
12	48384	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 7
13	2977792	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 727
14	?	?
15	?	?

2. Smallest palindromes with exactly n prime factors
    (counted with multiplicity).

Refer to sequence A076886 [ November 25, 2002 ]

by Shyam Sunder Gupta (extended by Robert G. Wilson v).

n	palindrome	prime factors
0	1	-
1	2	2
2	4	2 * 2
3	8	2 * 2 * 2
4	88	2 * 2 * 2 * 11
5	252	2 * 2 * 3 * 3 * 7
6	2772	2 * 2 * 3 * 3 * 7 * 11
7	27872	2 * 2 * 2 * 2 * 2 * 13 * 67
8	2112	2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
9	4224	2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
10	8448	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
11	44544	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 29
12	48384	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 7
13	2977792	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 727
14	27011072	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 11 * 11 * 109
15	405504	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 11
16	4091904	212 * 33 * 37
17	441606144	213 * 3 * 7 * 17 * 151
18	405909504	212 * 32 * 7 * 112 * 13
19	886898688	216 * 3 * 13 * 347
20	677707776	216 * 33 * 383
21	4285005824	219 * 11 * 743
22	276486684672	218 * 3 * 11 * 31 * 1031
23	21128282112	219 * 3 * 7 * 19 * 101
24	633498894336	217 * 3 * 74 * 11 * 61
25	2701312131072	222 * 3 * 19 * 11299
26	8691508051968	222 * 3 * 7 * 101 * 977

3. Smallest palindromes with exactly n distinct prime factors.

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046399

From index nr 10 to 13 from Donovan Johnson

From index nr 14 to 35 from Giovanni Resta.

Entries copied dd. 13 dec 2020.

n	palindrome	distinct prime factors
0	1	-
1	2	2
2	6	2 * 3
3	66	2 * 3 * 11
4	858	2 * 3 * 11 * 13
5	6006	2 * 3 * 7 * 11 * 13
6	222222	2 * 3 * 7 * 11 * 13 * 37
7	22444422	2 * 3 * 7 * 11 * 13 * 37 * 101
8	244868442	2 * 3 * 7 * 13 * 17 * 23 * 31 * 37
9	6434774346	2 * 3 * 7 * 11 * 13 * 17 * 19 * 31 * 107
10	438024420834	2 * 3 * 7 * 11 * 13 * 17 * 19 * 43 * 59 × 89
11	50146955964105	3 * 5 * 7 * 11 * 13 * 17 * 19 * 37 * 43 * 67 * 97
12	2415957997595142	2 * 3 * 7 * 11 * 13 * 17 * 23 * 29 * 31 * 37 * 157 * 197
13	495677121121776594	2 * 3 * 7 * 11 * 13 * 17 * 23 * 31 * 43 * 71 * 73 * 137 * 223
14	22181673755737618122	2 * 3 * 7 * 11 * 13 * 17 * 19 * 23 * 31 * 37 * 83 * 101 * 149 * 347
15	8789941164994611499878	2 * 3 * 7 * 11 * 13 * 17 * 19 * 23 * 37 * 41 * 79 * 83 * 173 * 239 * 479

4. Smallest palindromes with exactly n distinct palindromic prime factors.

Variation on the theme by Amarnath Murthy see A087331

n	palindrome	distinct palindromic prime factors
0	1	-
1	2	2
2	6	2 * 3
3	66	2 * 3 * 11
4	6666	2 * 3 * 11 * 101
5	?	?
6	?	?
7	?	?

5. Palindromes divided by the palindromic sum of their prime factors
    (counted with multiplicity) is a palindrome.

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046362

From index nr 5 to 12 from Donovan Johnson

From index nr 13 to 39 from Giovanni Resta.

Entries copied dd. 12 dec 2020.

n	palindrome	prime factors	palindromic sum	palindromic quotient
1	4	2 * 2	4	1
2	1345431	3 * 17 * 23 * 31 * 37 (all distinct)	111	12121
3	12333321	3 * 3 * 7 * 11 * 13 * 37 * 37	111	111111
4	1425665241	3 * 3 * 7 * 7 * 11 * 13 * 13 * 37 * 47	141	10111101
5	12026255262021	3 * 3 * 7 * 11 * 13 * 113 * 127 * 167 * 557	1001	12014241021
6	35686666668653	7 * 11 * 11 * 11 * 13 * 31 * 101 * 139 * 677	1001	35651015653
7	40181955918104	2 * 2 * 2 * 7 * 7 * 7 * 7 * 11 * 13 * 101 * 241 * 601	1001	40141814104
8	128449444944821	7 * 11 * 11 * 13 * 19 * 31 * 167 * 233 * 509	1001	128321123821
9	146426787624641	73 * 137 * 1433 * 1487 * 6871 (all distinct)	10001	14641214641
10	651751101157156	2 * 2 * 7 * 11 * 11 * 13 * 17 * 43 * 197 * 211 * 487	1001	651100001156
11	1734573883754371	11 * 31 * 59 * 73 * 137 * 991 * 8699 (all distinct)	10001	173440044371
12	2313421221243132	2 * 2 * 3 * 3 * 7 * 11 * 13 * 17 * 29 * 37 * 47 * 103 * 727	1001	2311110111132
13	6858874224788586	2 * 3 * 7 * 11 * 11 * 11 * 13 * 41 * 47 * 71 * 101 * 683	1001	6852022202586
14	9889882002889889	7 * 11 * 13 * 71 * 101 * 149 * 149 * 229 * 271	1001	9880002000889
15	12446686768664421	3 * 3 * 3 * 3 * 7 * 7 * 13 * 13 * 37 * 37 * 37 * 37 * 9901	10101	1232223222321
16	23265253035256232	2 * 2 * 2 * 7 * 7 * 11 * 11 * 13 * 13 * 29 * 29 * 47 * 101 * 727	1001	23242011024232
17	30153338083335103	7 * 13 * 73 * 137 * 2663 * 3119 * 3989 (all distinct)	10001	3015032305103
18	35303631013630353	3 * 11 * 11 * 59 * 73 * 101 * 137 * 173 * 9433	10001	3530010100353
19	51169348884396115	5 * 31 * 73 * 137 * 2713 * 3041 * 4001 (all distinct)	10001	5116423246115
20	74898935253989847	3 * 3 * 3 * 7 * 11 * 11 * 13 * 37 * 293 * 4243 * 5477	10101	7415002005147
21	113274969969472311	3 * 7 * 7 * 11 * 13 * 13 * 19 * 37 * 79 * 821 * 9091	10101	11214233241211
22	113754377773457311	11 * 73 * 79 * 137 * 659 * 3761 * 5281 (all distinct)	10001	11374300347311
23	211126995599621112	2 * 2 * 2 * 3 * 3 * 11 * 37 * 73 * 89 * 137 * 929 * 8713	10001	21110588501112
24	220471776677174022	2 * 3 * 3 * 7 * 7 * 11 * 13 * 13 * 13 * 17 * 41 * 101 * 349 * 421	1001	220251525152022
25	276628771177826672	2 * 2 * 2 * 2 * 11 * 17 * 73 * 137 * 449 * 3623 * 5683	10001	27660111106672
26	364465626626564463	3 * 7 * 11 * 13 * 37 * 43 * 103 * 103 * 131 * 131 * 419	1001	364101525101463
27	412268889988862214	2 * 3 * 7 * 11 * 73 * 103 * 137 * 151 * 647 * 8867 (all distinct)	10001	41222766722214
28	422752764467257224	2 * 2 * 2 * 3 * 7 * 11 * 13 * 19 * 19 * 37 * 83 * 149 * 307 * 347	1001	422330434033224
29	1558984743474898551	3 * 7 * 19 * 41 * 79 * 271 * 601 * 797 * 9293 (all distinct)	11111	140310030013041
30	2054708880888074502	2 * 3 * 7 * 13 * 37 * 73 * 137 * 1069 * 1291 * 7369 (all distinct)	10001	205450343054502
31	2246467776777646422	2 * 3 * 7 * 7 * 11 * 11 * 13 * 23 * 41 * 211 * 3313 * 7369	11011	204020323020402
32	2384883612163884832	2 * 2 * 2 * 2 * 2 * 7 * 7 * 11 * 11 * 13 * 23 * 43 * 89 * 97 * 269 * 421	1001	2382501111052832
33	3065376648466735603	7 * 11 * 11 * 13 * 53 * 79 * 101 * 103 * 113 * 163 * 347	1001	3062314334132603
34	3403712644462173043	7 * 11 * 11 * 13 * 41 * 89 * 97 * 107 * 151 * 191 * 283	1001	3400312332130043
35	5353666926296663535	3 * 5 * 11 * 11 * 31 * 73 * 73 * 101 * 137 * 137 * 9419	10001	535313161313535
36	5636156362636516365	3 * 5 * 11 * 11 * 11 * 29 * 9091 * 13921 * 76919	100001	56361000016365
37	5726821320231286275	3 * 3 * 5 * 5 * 7 * 11 * 11 * 13 * 37 * 79 * 101 * 787 * 9949	11011	520100020001025
38	7521979694969791257	3 * 19 * 37 * 53 * 73 * 137 * 419 * 2311 * 6949 (all distinct)	10001	752122757221257
39	8654288744478824568	2 * 2 * 2 * 3 * 3 * 3 * 11 * 11 * 47 * 9091 * 9533 * 81293	100001	86542022024568

6. Palindromes divisible by the palindromic sum of their prime factors
    (counted with multiplicity).

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046360

From index nr 3 on the terms are taken from the OEIS sequence dd. 13 dec 2020.

n	palindrome	palindromic prime factors	palindromic sum	quotient
1	4	2 * 2	4	1
2	5445	3 * 3 * 5 * 11 * 11	33	165
3	234432	2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 11 * 37	66	3552
4	585585	3 * 3 * 5 * 7 * 11 * 13 * 13	55	10647
5	888888	2 * 2 * 2 * 3 * 7 * 11 * 13 * 37	77	11544
6	951159	3 * 11 * 19 * 37 * 41 (all distinct)	111	8569
7	999999	3 * 3 * 3 * 7 * 11 * 13 * 37	77	12987
8	1345431	3 * 17 * 23 * 31 * 37 (all distinct)	111	12121
9	2554552	2 * 2 * 2 * 7 * 11 * 11 * 13 * 29	77	33176
10	4620264	2 * 2 * 2 * 3 * 11 * 11 * 37 * 43	111	41624
11	5842485	3 * 3 * 5 * 11 * 11 * 29 * 37	99	59015
12	6151516	2 * 2 * 7 * 19 * 31 * 373	434	14174
13	9704079	3 * 3 * 7 * 11 * 11 * 19 * 67	121	80199
14	12333321	3 * 3 * 7 * 11 * 13 * 37 * 37	111	111111
15	40199104	2 * 2 * 2 * 2 * 2 * 2 * 11 * 11 * 29 * 179	242	166112
16	42266224	2 * 2 * 2 * 2 * 7 * 7 * 11 * 13 * 13 * 29	88	480298
17	42666624	2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 7 * 11 * 13 * 37	88	484848
18	44088044	2 * 2 * 7 * 7 * 11 * 11 * 11 * 13 * 13	77	572572
19	46355364	2 * 2 * 3 * 3 * 11 * 19 * 61 * 101	202	229482
20	63599536	2 * 2 * 2 * 2 * 7 * 11 * 11 * 13 * 19 * 19	88	722722
21	63633636	2 * 2 * 3 * 3 * 11 * 37 * 43 * 101	202	315018
22	420838024	2 * 2 * 2 * 29 * 61 * 131 * 227	454	926956
23	424848424	2 * 2 * 2 * 7 * 7 * 11 * 11 * 13 * 13 * 53	121	3511144
24	631949136	2 * 2 * 2 * 2 * 3 * 7 * 13 * 13 * 31 * 359	434	1456104
25	649363946	2 * 7 * 11 * 83 * 101 * 503 (all distinct)	707	918478

7. Palindromes divisible by the palindromic sum of their exclusive palindromic prime factors.
    Nonpalindromic prime factors may not occur.
    (counted with multiplicity).

n	palindrome	palindromic prime factors	palindromic sum	quotient
1	4	2 * 2	4	1
2	5445	3 * 3 * 5 * 11 * 11	33	165
3	?	?	?	?
4	?	?	?	?

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E-mail address : pdg@worldofnumbers.com

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